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%\title{NRC Staff Capacity Building: \\
% Micromechanical Origins of ElastoPlasticity }
\title{On Earthquake Soil Structure Interaction \\
Modeling and Simulation}
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
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\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
{N.~Tafazzoli, J.A.~Abell~Mena, B.~Kamrani,
C.G.~Jeong, K.~Watanabe, B.~Aldridge, J.~Anderson, F.~Pisan{\`o}, M.~Martinelli,
K.~Sett, \\ ~ \\
B.~Jeremi{\'c}}
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\institute[\pgfuseimage{universitylogo}\hspace*{0.1truecm}\pgfuseimage{lbnllogo}] % (optional, but mostly needed)
{ Professor, University of California, Davis\\
% and\\
Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
%  Use the \inst command only if there are several affiliations.
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\date[] % (optional, should be abbreviation of conference name)
{\small UCD Geotech. Seminar, May, 2013}
\subject{}
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\section{Motivation}
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% \begin{frame}
% \frametitle{}
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% \begin{itemize}
% %\vspace*{0.3cm}
% \item
%
% \item
% \end{itemize}
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\subsection{Motivation}
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\begin{frame}
\frametitle{The Problem}
\begin{itemize}
\item Seismic response of Nuclear Power Plants
\vspace*{0.1cm}
\item 3D, inclined seismic motions consisting of body and surface waves
\vspace*{0.1cm}
\item Inelastic (elastic, damage, plastic behavior of materials: soil, rock,
concrete, steel, rubber, etc.)
\vspace*{0.1cm}
\item Full coupling of pore fluids (in soil and rock) with soil/rock skeleton
\vspace*{0.1cm}
\item Buoyant effects (foundations below water table)
\vspace*{0.1cm}
\item Uncertainty in seismic sources, path, soil/rock response and structural
response
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Solution}
\begin{itemize}
%\vspace*{0.3cm}
\item {\bf Physics based modeling and simulation} of seismic behavior of
soilstructure systems (NPP structures, components and systems)
\vspace*{0.1cm}
\item Development and use of {\bf high fidelity} time domain,
nonlinear numerical models,
in {\bf deterministic} and {\bf probabilistic} spaces
\vspace*{0.1cm}
\item Accurate following of the {\bf flow of seismic
energy} (input and dissipation) within soilstructure NPP system
\vspace*{0.1cm}
\item {\bf Directing}, in space and time, with {\bf high (known)
confidence}, seismic energy flow in the soilfoundationstructure system
%\vspace*{0.1cm}
% \item {\bf Education} for researchers, professional practice.
\end{itemize}
\end{frame}
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\section{Challenges}
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\subsection{Uncertainty in Modeling Ground Motions}
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\begin{frame}
\frametitle{Modeling Uncertainty}
\begin{itemize}
%\vspace*{0.3cm}
\item Simplified (or inadequate/wrong) modeling: important features are
missed (seismic ground motions, etc.)
\vspace*{0.2cm}
\item Introduction of uncertainty and (unknown) lack of accuracy in results due
to use of unverified simulation tools (software quality, etc.)
\vspace*{0.2cm}
\item Introduction of uncertainty and (unknown) lack of accuracy in results due
to use of unvalidated models (due to lack of quality validation experiments)
% (still missing data, experiments under
% uncertainty, for more see below)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Complexity of and Uncertainty in Ground Motions}
\begin{itemize}
%\vspace*{0.3cm}
\item 6D (3 translations, 3 rotations)
\vspace*{0.3cm}
\item Vertical motions usually neglected
\vspace*{0.3cm}
\item Rotational components usually not measured and neglected
\vspace*{0.3cm}
\item Lack of models for such 6D motions (from measured data))
\vspace*{0.3cm}
\item Sources of uncertainties in ground motions (Source, Path (rock), soil (rock))
\end{itemize}
\end{frame}
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\subsection{Uncertainty in Modeling Material}
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\begin{frame}
\frametitle{Material Behavior Inherently Uncertain}
%\begin{itemize}
%\vspace*{0.5cm}
%\item
%Material behavior is inherently uncertain (concrete, metals, soil, rock,
%bone, foam, powder etc.)
\begin{itemize}
\vspace*{0.5cm}
\item Spatial \\
variability
\vspace*{0.5cm}
\item Pointwise \\
uncertainty, \\
testing \\
error, \\
transformation \\
error
\end{itemize}
% \vspace*{0.5cm}
% \item Failure mechanisms related to spatial variability (strain localization and
% bifurcation of response)
%
% \vspace*{0.5cm}
% \item Inverse problems
%
% \begin{itemize}
%
% \item New material design, ({\it pointwise})
%
% \item Solid and/or structure design (or retrofits), ({\it spatial})
%
% \end{itemize}
%\end{itemize}
\vspace*{5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Soil Uncertainties and Quantification}
\begin{itemize}
%
%\vspace*{0.5cm}
\item Natural variability of soil deposit (Fenton 1999)
\begin{itemize}
\item Function of soil formation process
\end{itemize}
%
\vspace*{0.2cm}
\item Testing error (Stokoe et al. 2004)
\begin{itemize}
\item Imperfection of instruments
\item Error in methods to register quantities
\end{itemize}
%
\vspace*{0.2cm}
\item Transformation error (Phoon and Kulhawy 1999)
\begin{itemize}
\item Correlation by empirical data fitting (e.g. CPT data $\rightarrow$ friction angle etc.)
\end{itemize}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{SPT Based Determination of Shear Strength}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation of SPT $N$value $\rightarrow$ undrained shear
strength, $s_u$ (cf. Phoon and Kulhawy (1999B)
Histogram of the residual
(w.r.t the deterministic transformation
equation) undrained strength,
along with fitted probability density function
(Pearson IV)
\end{frame}
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\begin{frame}
\frametitle{SPT Based Determination of Young's Modulus}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation of SPT $N$value $\rightarrow$ 1D Young's modulus, $E$ (cf. Phoon and Kulhawy (1999B))
Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function
\end{frame}
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\subsection{Errors in Scientific Software}
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\begin{frame}
\frametitle{Errors in Scientific Software: The T Experiments}
\begin{itemize}
% \vspace*{0.2truecm}
\item Les Hatton, Kingston University (formerly of Oakwood Comp. Assoc.)
\vspace*{0.1truecm}
\item "Extensive tests showed that many software codes widely used in science
and engineering are not as accurate as we would like to think."
\vspace*{0.1truecm}
\item "Better software engineering practices would help solve this problem,"
\vspace*{0.1truecm}
\item "Realizing that the problem exists is an important first step."
\vspace*{0.1truecm}
\item Large experiment over 4 years measuring faults (T1) and failures (T2)
of scientific and engineering codes
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The T2 Experiments}
\begin{itemize}
\item Specific application area: seismic data processing (inverse analysis)
\vspace*{0.2truecm}
\item Echo sounding of underground and reconstructing "images" of
subsurface geological structure
\vspace*{0.2truecm}
\item Nine mature packages, using {\bf same algorithms}, on a {\bf same data set}!
\vspace*{0.2truecm}
\item 14 primary calibration points for results check
\vspace*{0.2truecm}
\item Results "fascinating and disturbing"
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{T2: Disagreement at Calibration Points}
\begin{figure}[!h]
\begin{center}
\hspace*{1.5cm}
%\vspace*{2.5cm}
{\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/GheoMat/VandV_01/T2_01.jpg}}
\hspace*{1.5cm}
%\vspace*{5.0cm}
\end{center}
\end{figure}
% \begin{itemize}
%
%
%
% \end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% \begin{frame}
% \frametitle{T2: Stage 14, Interpretation of Data }
%
%
%
%
% \begin{figure}[!h]
% \begin{center}
% %\vspace*{2.5cm}
% \vspace*{1.0cm}
% \hspace*{1.5cm}
% {\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/GheoMat/VandV_01/T2_02.jpg}}
% \hspace*{1.5cm}
% \vspace*{1.5cm}
% %\vspace*{5.0cm}
% \end{center}
% \end{figure}
%
% %
% % % \begin{itemize}
% % %
% % %
% %
% % \end{itemize}
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\section{ESSI Simulator System}
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\subsection{ESSI Simulator System}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI Simulator System}
\begin{itemize}
\item {\bf The ESSIProgram} is a 3D, nonlinear, time domain,
parallel finite element program specifically developed for
HiFi modeling and simulation of Earthquake Soil/Rock Structure
Interaction problems for NPPs on ESSIComputer. \
%The NRC ESSI Program is based on
%a number of public domain numerical libraries developed at UCD as well as those
%available on the web, that are compiled and linked together to form the
%executable program (NRCESSIProgram). Significant effort is devoted to development
%of verification and validation procedures, as well as on development of
%extensive documentation. NRCESSIProgram is in public domain and is licensed
%through the Lesser GPL.
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The ESSIComputer} is a distributed memory
parallel computer, a cluster of clusters with multiple performance
processors and multiple performance networks.
%Compute nodes are Shared Memory Parallel
%(SMP) computers, that are connected, using high speed network(s), into a
%Distributed Memory Parallel (DMP) computer.
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The ESSINotes} represent a hypertext
documentation system
%(Theory and Formulation, Software and Hardware, Verification and Validation, and
%Case Studies and Practical Examples)
detailing modeling and simulation of NPP ESSI
problems.
%
%the
%NRCESSIProgram code API (application Programming Interface) and DSLs (Domain
%Specific Language).
%%NRCESSINotes, developed by Boris Jeremic and collaborators, are in public
%domain
%%and are licensed under a Creative Commons AttributionShareAlike 3.0 Unported
%%License.
%
%\vspace*{0.3cm}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI Simulator Program}
\begin{itemize}
%\vspace*{0.2cm}
\item Based on a Collection of Useful Libraries (modular, portable)
\vspace*{0.1cm}
\item Library centric software design
\vspace*{0.1cm}
\item Various public domain licenses (GPL, LGPL, BSD, CC)
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item Verification (extensive) and Validation (not much)
\vspace*{0.1cm}
\item Program documentation (part of ESSI Notes)
\vspace*{0.1cm}
\item Target users: USNRC staff, CNSC staff, IAEA, LBNL, INL, DOE,
professional practice collaborators, expert users
%\item Sources will be available through
%{\bf
%\url{http://nrcessisimulator.info}}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Collection of Useful Libraries (Modeling Part)}
\begin{itemize}
\vspace*{0.2cm}
\item Template3DEP libraries for elastic and elasticplastic
computations (UCD, CC)
\vspace*{0.2cm}
\item FEMTools finite element libraries provide
finite elements (solids,
beams, shells, contacts/isolators, seismic input) (UCD, UCB, CU, CC)
\vspace*{0.2cm}
\item Loading, staged, self weight, service loads, seismic loads
(the Domain Reduction Method, analytic input
(incoming/outgoing) of 3D, inclined, uncorrelated seismic motions)
(UCD, CC)
\vspace*{0.2cm}
\item Domain Specific Language for input (UCD, CC)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Collection of Useful Libraries (Simulation Part)}
\begin{itemize}
\vspace*{0.12cm}
\item Plastic Domain Decomposition (PDD) for parallel computing (UCD, CC)
\vspace*{0.12cm}
\item PETSc (ANL, GPLlike) and UMFPACK (UF, GPL) solvers
\vspace*{0.12cm}
\item Modified OpenSees Services (MOSS) for managing the finite
element domain (UCD, CC; UCB, GPL?)
\vspace*{0.12cm}
\item nDarray (UCD, CC), LTensor (CIMEC, GPL),
BLAS (UTK, GPL) for lower level
computational tasks,
\vspace*{0.12cm}
\item Message Passing Interface (MPI, openMPI, new BSD license)
\end{itemize}
\end{frame}
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\begin{frame}[fragile]
\frametitle{ESSI Simulator Computer}
A distributed memory parallel (DMP) computer
designed for high performance,
parallel finite element simulations
\begin{itemize}
%\vspace*{0.1cm}
\item Multiple performance CPUs \\
and Networks
%\vspace*{0.1cm}
\item Most costperformance \\
effective
%\vspace*{0.1cm}
\item Source compatibility with \\
any DMP supercomputers
%\vspace*{0.1cm}
\item Current systems: 208CPUs, \\
and 48CPUs (+64) and \\
96CPUs (8x5+2x16+24)...
%%\vspace*{0.1cm}
% \item Near future: 784 CPUs
\end{itemize}
\vspace*{4.5cm}
\begin{flushright}
%\hspace*{0.5cm}
\includegraphics[width=5.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2607.JPG}
%\includegraphics[width=6.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2609.JPG}
%\includegraphics[width=8.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2611.JPG}
\end{flushright}
\end{frame}
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% %\begin{frame}[fragile]
% % \frametitle{NRC ESSI Simulator Version December 2010}
% %
% %
% %\begin{itemize}
% %\item Operating System: Linux Fedora Core 14.
% %
% %\item Kernel: \verb2.6.35.1074.fc14.x86_64
% %
% %\item Compute Nodes (two):
% %
% % \begin{itemize}
% % \item CPU: 2 $\times$ Intel Xeon E5620
% % Westmere 2.4 GHz Quad Core (8 threads)
% %
% % \item RAM: 6 $\times$ 4GB DDR3 1333 MHz ECC/Registered Memory (24GB
% % Total Memory)
% %
% % \item Disk: 8 $\times$ 500 GB Seagate Constellation ES 3.5" SATA/300
% % (Linux Software RAID10)
% %
% % \end{itemize}
% %
% %\item Network: single GigaBit
% %\end{itemize}
% %
% %
% %\end{frame}
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\begin{frame}[fragile]
\frametitle{ESSI Computer Version April 2012}
Operating System: Ubuntu
Kernel: 3.2
{\bf Controller:} 1 node + {\bf Compute:} 8 Nodes
\begin{itemize}
\item CPU: 2 x 12 cores Opteron 6234 = 24 cores
\item RAM: 32GB (8 x 4GB)
\item NICs:
\begin{itemize}
\item GigaBit: Intel 82576 (Controller)
\item InfiniBand: ConnectX2 QDR IB 40Gb/s (Controller+Compute)
\end{itemize}
\item Disk: 8 $\times$ 2TB Toshiba MK2002TSKB (Controller)
\item Disk: 1TB Toshiba MK1002TSKB (Compute)
\end{itemize}
Network (dual):
\begin{itemize}
\item GigaBit: HP ProCurve Switch 181048G 48 Port
\item InfiniBand:: Mellanox MIS5030Q1SFCA 36port QDR
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{ESSI Simulator Notes}
\begin{itemize}
\item A hypertext documentation system describing in detail modeling and
simulations of NPP ESSI problems
\begin{itemize}
\item Theoretical and Computational Formulations (FEM, ELPL, Static
and Dynamic solution, Parallel Computing)
\item Software and Hardware Platform Design (OO Design, Library centric
design, API, DSL, Software Build Process, Hardware Platform)
\item Verification and Validation (code V, Components V, Static and
Dynamic V, Wave Propagation V)
\item Application to Practical Nuclear Power Plant Earthquake
Soil/Rock Structure Interaction Problems (ESSI with 3D, inclined,
uncorrelated seismic waves, ESSI with foundation slip, Isolators)
\end{itemize}
\end{itemize}
\end{frame}
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\subsection{Modeling and Simulation}
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\begin{frame}
% \frametitle{Seismic Energy Dissipation for \underline{Soil}FoundationStructure Systems}
\frametitle{High Fidelity Modeling}
% \frametitle{Seismic Energy Dissipation for
% \underline{Soil}FoundationStructure Systems}
\begin{itemize}
\item Seismic energy influx, {body and surface waves, 3D, inclined}
% $E_{flux} = \rho A c \int_0^t \dot{u}_i^2 dt$ (Aki \& Richards)
\vspace*{0.1cm}
\item Mechanical dissipation outside of SSI domain:
\begin{itemize}
\item {Radiation} of reflected waves
\item {Radiation} of oscillating SSI system
\end{itemize}
\vspace*{0.1cm}
\item Mechanical dissipation inside SSI domain:
\begin{itemize}
\item {Plasticity} of soil/rock subdomain
\item {Viscous coupling} of porous solid with pore fluid (air,
water)
\item {Plasticity} and viscosity of foundation  soil/rock contact
\item Plasticity/damage of the structure
\item Viscous coupling of structure/foundation with fluids
% \item potential and kinetic energy
% \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\end{itemize}
\vspace*{0.1cm}
% \item Numerical energy dissipation (numerical damping/production and period errors)
% \item Numerical energy dissipation (damping/production)
\item Numerical energy dissipation/production
\end{itemize}
%
\end{frame}
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\begin{frame}
\frametitle{High Performance, Parallel Computing}
\begin{itemize}
\item The ESSI Program can be used in both sequential and
parallel modes
\vspace*{0.2cm}
\item For high fidelity models, parallel is really the only option
\vspace*{0.2cm}
\item High performance, parallel computing using
Plastic Domain Decomposition
Method
\vspace*{0.2cm}
\item Developed for multiple/variable capability CPUs and
networks (DMP and
SMPs)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Finite Elements}
\begin{itemize}
\item Linear and nonlinear truss element
\item Linear and nonlinear beam (disp. based), variable BC el.
\item Linear shell Triangle and Quad with drilling DOFs
%\item Linear and nonlinear thick shell (bricks)
\item Single phase solid bricks (8, 20, 27, 820, 827 nodes)
\item Two phase (fully coupled, porous solid, pore fluid) solid bricks (8 and
20 node: $upU$, $up$)
\item Dry friction slip and gap element
\item Saturated gap and slip element
\item Seismic isolator (latex rubber, neoprene rubber, rubber with lead core,
friction pendulum)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Material Models for Solids and Structures}
\begin{itemize}
\item Small deformation elastic: linear, nonlinear isotropic, cross anisotropic
\vspace*{0.3cm}
\item Small deformation elasticPlastic: von Mises,
DruckerPrager, CamClay, Rounded MohrCoulomb, Parabolic Leon,
SaniSand2004, SaniSand2008, SaniClay, Pisan{\`o}; Gens normal contact and
Coulomb shear contact model; 1D concrete and steel models
%\vspace*{0.3cm}
% \item Isotropic and kinematic (translational and rotational) hardening
\vspace*{0.3cm}
\item Large deformation elastic and elasticplastic: Ogden, neoHookean,
MooneyRivlin, Logarithmic, SimoPister, von Mises, DruckerPrager
\end{itemize}
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Earthquake Ground Motions}
Realistic earthquake ground motions
\begin{itemize}
\vspace*{0.1cm}
\item Body: P and S waves
\vspace*{0.1cm}
\item Surface: Rayleigh, Love waves, etc.
\vspace*{0.1cm}
\item Lack of correlation (incoherence)
\vspace*{0.1cm}
\item Inclined waves
\vspace*{0.1cm}
\item 3D waves
%\vspace*{0.1cm}
% \item Earthquake energy dissipation
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Body (P, S) and Surface (Rayleigh, Love) Waves}
\vspace*{0.3cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=2.5cm, angle=45]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/P_body_wave.jpeg}
\includegraphics[width=2.5cm, angle=45]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/S_body_wave.jpeg}
\vspace*{0.7cm}
\\
\includegraphics[width=3cm]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/Rayleigh_surface_wave.jpeg}
\includegraphics[width=3cm]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/Love_surface_wave.jpeg}
%\caption{\label{Love_surface_wave} Visualization of propagation of a Love
%surface seismic wave (illustrations are from MTU web site).}
\end{center}
\end{figure}
\end{frame}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Spatial Variability (Incoherence, Lack of Correlation)}
% 
%  Incoherence $\rightarrow$ frequency domain
% 
%  \vspace*{0.2cm}
% 
%  Lack of Correlation $\rightarrow$ time domain
% 
% 
%  \vspace*{0.5cm}
% 
%  \begin{itemize}
%  \item Attenuation effects
%  \item Wave passage effects
%  \item Extended source effects
%  \item Scattering effects
%  \item Variable seismic energy dissipation
%  \end{itemize}
% 
%  %\begin{figure}[!htb]
%  %\begin{center}
%  \vspace*{3.5cm}
%  \hspace*{5.5cm}
%  \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
%  %\caption{\label{LC} Four main sources contributing to the lack of correlation of
%  %seismic waves as measured at two observation points.}
%  %\end{center}
%  %\end{figure}
%  %
%  %A number of models available (Abrahamson...)
%  %
%  \end{frame}
% 
% 
%  % out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % out \begin{frame}
% out \frametitle{Attenuation Effects}
% out
% out
% out Responsible for change in amplitude and phase of seismic motions
% out due to the distance between observation points and losses (damping, energy dissipation) that
% out seismic wave experiences along that distance. This is a significant source of lack of correlation
% out for long structures (bridges), however for NPPs it is not of much significance.
% out
% out
% out %\begin{figure}[!htb]
% out \begin{center}
% out %\vspace*{2cm}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out %\caption{\label{LC} Four main sources contributing to the lack of correlation of
% out %seismic waves as measured at two observation points.}
% out \end{center}
% out %\end{figure}
% out
% out
% out
% out \end{frame}
% out
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Wave Passage Effects}
% out
% out Contribute to lack of correlation due to difference in
% out recorded wave field at two location points as the (surface) wave travels,
% out propagates from the first to second point.
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out
% out \end{frame}
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Extended Source Effects}
% out
% out Contribute to lack of correlation by creating a complex wave source
% out field, as the fault ruptures, rupture propagates and generate seismic sources along the fault.
% out Seismic energy is thus emitted from different points (along the rupturing fault) and will have
% out different travel path and timing as it makes it observation points.
% out
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out \end{frame}
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Scattering Effects}
% out
% out Responsible to lack of correlation by creating a
% out scattered wave field.
% out Scattering is due to (unknown or not known enough) subsurface geologic features
% out that contribute to (elastic) modification of the wave field.
% out
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out
% out
% out \end{frame}
% out
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Variable Seismic Energy Dissipation}
% out
% out Contribute to variability of seismic motions by bending seismic waves as they
% out pass through inelastic soil/rock.
% out Variable seismic energy dissipation is due to (unknown or not known enough)
% out subsurface geologic features that contribute to (inelastic, elasticplastic)
% out modification of the wave field.
% out
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out
% out
% out
% out \end{frame}
% out
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Modeling Lack of Correlation (Incoherence)}
% out
% out \begin{itemize}
% out
% out \item A number of models available (Abrahamson...)
% out
% out \item Most of the models (all) are based on a (very) limited data set from Lotung, Pinyon Flat...
% out
% out \item Most of the models (all) are based on hard rock data ($V_s > 2600$m/s)
% out
% out \item Most of the models (all) can produce statistically significant number of
% out motions, yet only few are used (destroying the model statistical assumptions)
% out
% out
% out \item Ergodic assumption must be made in order to extrapolate those models
% out (data) to other parts of the USA (world)
% out
% out %\item Extrapolations can be (are) dangerous
% out
% out
% out \end{itemize}
% out
% out
% out \end{frame}
% out
% out
% out
% out
% out
% out
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Seismic Input}
% 
% 
%  %\begin{itemize}
% 
%  % \item
%  The Domain Reduction Method \\
%  (Bielak et al.): \\
%  The effective force $P^{eff}$ \\
%  is a dynamically consistent \\
%  replacement for the dynamic \\
%  source forces $P_{e}$
% 
%  % \end{itemize}
% 
%  \begin{eqnarray}
%  P^{eff} = \left\{\begin{array}{c} P^{eff}_i \\ P^{eff}_b \\ P^{eff}_e \end{array}\right\}
%  = \left\{\begin{array}{c} 0 \\ M^{\Omega+}_{be} \ddot{u}^0_eK^{\Omega+}_{be}u^0_e
%  \\ M^{\Omega+}_{eb}\ddot{u}^0_b+K^{\Omega+}_{eb}u^0_b\end{array}\right\}
%  \nonumber
%  \label{DRMeq09}
%  \end{eqnarray}
%  %
% 
%  \begin{figure}[!h]
%  \begin{flushright}
%  %\vspace*{0.50cm}
%  %\begin{center}
%  %\hspace*{1cm}
%  \vspace*{6.90cm}
%  {\includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2010/NRCLBLProjectReviewMeeting_21_22_Sept_2010/DRM05NPP.pdf}}
%  %\vspace*{5.50cm}
%  %\hspace*{1cm}
%  %\vspace*{2.50cm}
%  %\end{center}
%  %\vspace*{0.3cm}
%  \end{flushright}
%  \end{figure}
% 
% 
% 
% 
%  \end{frame}
% 
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{DRM}
\begin{itemize}
%\vspace*{0.2cm}
\item Seismic forces $P_e$ replaced by $P^{eff}$
%\vspace*{0.2cm}
\item $P^{eff}$ applied only to a single \\
layer of elements next to $\Gamma$.
%\vspace*{0.2cm}
\item The only outgoing waves are \\
from dynamics of the NPP
%\vspace*{0.2cm}
\item Material inside $\Omega$ \\
can be elasticplastic
\item All types of seismic waves\\
(body, surface...) are \\
properly modeled
% \item The only input wave field is the one for the nodes of this layer of elements.
\end{itemize}
\begin{figure}[!h]
\begin{flushright}
%\vspace*{0.50cm}
%\begin{center}
%\hspace*{1cm}
\vspace*{4.50cm}
{\includegraphics[width=5.8cm]{/home/jeremic/tex/works/Conferences/2010/NRCLBLProjectReviewMeeting_21_22_Sept_2010/DRM05NPP.pdf}}
%\vspace*{5.50cm}
\hspace*{0.8cm}
%\vspace*{2.50cm}
%\end{center}
%\vspace*{0.3cm}
\end{flushright}
\end{figure}
\end{frame}
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\subsection{Verification and Validation Suite}
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\begin{frame}
\frametitle{Verification, Validation and Prediction}
\begin{itemize}
\item Verification: the process of determining that a model
implementation accurately represents the developer's conceptual description
and specification. Mathematics issue. {\it Verification provides evidence that the
model is solved correctly.}
\item Validation: The process of determining the degree to which a
model is accurate representation of the real world from the perspective of
the intended uses of the model. Physics issue. {\it Validation provides
evidence that the correct model is solved.}
\item Prediction: use of computational model to foretell the state of a
physical system under consideration under conditions for which the
computational model has not been validated
\end{itemize}
%
%\item Models available (some now, some later)
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Role of Verification and Validation}
\begin{figure}[!h]
\begin{center}
\hspace*{2cm}
{\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/ASME_V_and_V_symposium/presentetation/RoleVV_NEW_knowledge.pdf}}
{\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/VandV_ODEN.jpg}}
\hspace*{2cm}
\end{center}
\end{figure}
{Oberkampf et al. \hspace*{4cm} Oden et al.}
%
%\item Models available (some now, some later)
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Importance of V\&V}
\begin{itemize}
% \vspace*{2.0truecm}
\item V \& V procedures are the primary means of assessing accuracy in
modeling and computational simulations
\vspace*{0.5truecm}
\item V \& V procedures are the tools with which we build confidence and
credibility in modeling and computational simulations
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{V \& V for ESSI Modeling and Simulations}
\begin{itemize}
\vspace*{0.3cm}
\item Material modeling and simulation (elastic, elasticplastic...)
\vspace*{0.3cm}
\item Finite elements (solids, structural, special...)
\vspace*{0.3cm}
\item Solution advancement algorithms (static, dynamic...)
\vspace*{0.3cm}
\item Seismic input and radiation
\vspace*{0.3cm}
\item Finite element model verification
\end{itemize}
\end{frame}
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%
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\begin{frame}
\frametitle{Mesh Size Effects on Seismic Wave Propagation Modeling}
\begin{itemize}
\item Finite element mesh "filters out" \\
high frequencies
%\vspace*{0.2cm}
\item Usual rule of thumb: 1012 elements \\
needed per wave length
% (SASSI recommends only 5 ?!)
%
% \item Maximum grid spacing should not exceed
% $\Delta h \;\le\; {\lambda}/{10}\;=\;{v}/({10\,f_{max}})$
% where $v$ is the lowest wave velocity (shear, elasticplastic ?)
%
% \item Tests without and with numerical damping, for different element sizes
%
\item 1D wave propagation model
%\vspace*{0.2cm}
\item 3D finite elements (same in 3D)
%\vspace*{0.2cm}
\item Motions applied as displacements at the bottom
\end{itemize}
%\begin{figure}[H]
\vspace*{4.0cm}
\begin{flushright}
\includegraphics[width=0.7cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_21Nov2011/model01.pdf}
\end{flushright}
%\end{figure}
\vspace*{0.4cm}
\begin{small}
\begin{table}[!htbp]
\centering
% \begin{tabular}{ccccc}
\begin{tabular}{rm{2.6cm}m{1.5cm}m{1.8cm}m{2.3cm}}
\hline
case & model height [m] & $V_s$ [m/s] & El.size [m] & $f_{max}$ (10el) [Hz]\\
\hline
%\hline
3 & 1000 & 1000 & 10 & 10\\
%\hline
4 & 1000 & 1000 & 20 & 5\\
%\hline
6 & 1000 & 1000 & 50 & 2\\
\hline
% \begin{tabular}{m{1.5cm}cm{2.8cm}cm{2.8cm}cm{3.0cm}cm{4.0cm}c}
% \begin{tabularx}{\linewidth}{ccccc}
% \begin{tabular*}{0.75\textwidth}{@{\extracolsep{\fill}}ccccc}
\end{tabular}
% \end{tabularx}
\end{table}
\end{small}
\end{frame}
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\begin{frame}
\frametitle{Cases 3, 4, and 6, Ormsby Wavelet Input Motions}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/Input_Displacement.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Cases 3, 4, and 6, Surface Motions}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/displacement.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cases 3, 4, and 6, Input and Surface Motions, FFT}
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/FFT.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Seismic Body and Surface Waves}
\begin{itemize}
\item Both body (P, SV and SH) and surface (Rayleigh, Love, etc.) waves
are present
\vspace*{1mm}
\item Surface waves carry most seismic energy
\vspace*{1mm}
\item Analytic (Aki and Richards, Trifunac and Lee, Hisada et
al., fk, etc.) and numerically generated, 3D, inclined (plane) body and
surface waves are used in tests
\vspace*{1mm}
\item Seismic moment from a point source at $2$km depth used
\vspace*{1mm}
\item Stress drop at the source: Ricker and/or Ormsby wavelets
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Plane Wave Model}
\begin{figure}[H]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Seismic Source Mechanics}
\vspace*{0.5cm}
Stress drop, Ormsby wavelet
\vspace*{1cm}
\begin{figure}[H]
\begin{flushright}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/Seismic_source_moment_couple.pdf}
\end{flushright}
\end{figure}
\vspace*{1.9cm}
\hspace*{1cm}
\begin{figure}[H]
\begin{center}
\hspace*{0.4cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FaultModel_7seconds/xz_TimeHistory/3000_3000_x_displacement.pdf}
\hspace*{0.4cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FaultModel_7seconds/xz_FFT/3000_3000_x_displacement_FFT.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Middle (Structure Location) Plane, Top 2km}
\vspace*{0.4cm}
\begin{figure}[H]
\begin{flushright}
\includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model_MIDDLE.pdf}
\end{flushright}
\end{figure}
\vspace*{2.0cm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/Ormsby/middle_top2000/middle_acceleration_x.pdf}
\hspace*{0.5cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/Ormsby/middle_top2000/middle_acceleration_z.pdf}
\end{center}
\end{figure}
\vspace*{0.90cm}
{horizontal accelerations \hfill vertical accelerations}
\end{frame}
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\begin{frame}
\frametitle{Verification: Displacements, Top Middle Point }
% \begin{itemize}
% \item
% \end{itemize}
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\begin{figure}[!htbp]
\begin{center}
\begin{tabular}{ccc}
%\hline
(X)
&
(Z)
\\
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_disp_x.pdf}
&
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_disp_z.pdf}
&
\end{tabular}
%\caption{Comparison of displacements for top middle point using Ricker wave $(f=1Hz)$ as an input motion}
%\label{fig:ricker_acc}
\end{center}
\end{figure}
\end{frame}
%
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% \begin{frame}
% \frametitle{Verification: Accelerations, Top Middle Point }
% % \begin{itemize}
% % \item
% % \end{itemize}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{figure}[!htbp]
% \begin{center}
% \begin{tabular}{ccc}
% %\hline
% (X)
% &
% (Z)
% \\
% \includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_accel_x.pdf}
% &
% \includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_accel_z.pdf}
% &
% \end{tabular}
% %\caption{Comparison of accelerations for top middle point using Ricker wave $(f=1Hz)$ as an input motion}
% %\label{fig:ricker_acc}
% \end{center}
% \end{figure}
%
%
%
%
% \end{frame}
%
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\begin{frame}
\frametitle{Verification: Disp. and Acc., Out of DRM }
% \begin{itemize}
% \item
% \end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\begin{tabular}{ccc}
%\hline
Displacement
&
Acceleration
\\
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/10_40_disp_x.pdf}
&
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/10_40_accel_x.pdf}
&
\end{tabular}
%\caption{Displacement and acceleration time history for a point outside of DRM layer in (x) direction}
%\label{fig:out_ricker_disp}
\end{center}
\end{figure}
\end{frame}
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\section{Probabilistic Modeling}
\subsection{Uncertain (Geo) Materials}
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\begin{frame}
\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl constitutive equation
$\displaystyle \Delta \sigma_{ij} = D_{ijkl} \displaystyle \Delta \epsilon_{kl}$
%\begin{normalsize}
%
% \begin{equation}
% \nonumber
% \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
% \end{equation}
\begin{eqnarray}
\nonumber
D_{ijkl} = \left\{\begin{array}{ll}
%
D^{el}_{ijkl}
%
%
\;\;\; & \mbox{\large{~for elastic}} \\
%
\\
%
D^{el}_{ijkl}

\frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
{\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
\;\;\; & \mbox{\large{~for elasticplastic}}
%
\end{array} \right.
\end{eqnarray}
%\end{normalsize}
%\vspace{0.5cm}
% \item Nonlinear coupling in the ElPl modulus
\item What if all (any) material parameters are uncertain
\item PEP and SEPFEM methods for spatially variable and point uncertain material
% \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%
%
%
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t) \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
% \end{eqnarray}
%
\end{itemize}
%
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\end{frame}
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\begin{frame}
\frametitle{Probabilistic Stress Solution: \\ EulerianLagrangian form of FPK Equation}
%
%\begin{itemize}
% 3D
\begin{footnotesize}
\begin{eqnarray}
\nonumber
\lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
\left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial
\eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t),
\epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), E_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)
\vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
\vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), E_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau))
\vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
\end{eqnarray}
\end{footnotesize}
% 1D % 1D
% 1D \begin{footnotesize}
% 1D \begin{eqnarray}
% 1D \nonumber
% 1D &&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
% 1D  \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% 1D q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% 1D \nonumber
% 1D &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% 1D \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% 1D \nonumber
% 1D & & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% 1D \epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
% 1D \nonumber
% 1D &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% 1D \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% 1D \nonumber
% 1D & & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% 1D \epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
% 1D \nonumber
% 1D \end{eqnarray}
% 1D
% 1D \end{footnotesize}
\end{frame}
%
% \item 6 equations
%
% \item Complete description of 3D probabilistic stressstrain response
%
% \end{itemize}
%
%
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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation and (SEP)FEM}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma_{ij},t)}{\partial t}
=
\frac{\partial}{\partial\sigma_{ab}}
\left[N_{ab}^{(1)}P(\sigma_{ij},t)

\frac{\partial}{\partial \sigma_{cd}}
\left\{N_{abcd}^{(2)} P(\sigma_{ij},t)\right\} \right]
\end{equation}
%
\vspace*{0.1cm}
\item {\bf Complete} probabilistic description of response
\vspace*{0.1cm}
\item {\bf Secondorder exact} to covariance of time (exact mean and variance)
% 
%  \item Deterministic equation in probability density space
% 
%  \item Linear PDE in probability density space
%  $\rightarrow$ simplifies the numerical solution process
% 
%\item Applicable to any elasticplasticdamage material model (only coefficients $N_{ab}^{(1)}$
%and $N_{abcd}^{(2)}$ differ)
\vspace*{0.1cm}
\item Any uncertain FEM problem
(${\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
$)
with
\begin{itemize}
\item uncertain material parameters (stiffness matrix ${\bf K}$),
\item uncertain loading (load vector ${\bf F}$)
\end{itemize}
can be analyzed using PEP and SEPFEM to obtain PDFs of DOFs,
stress, strain...
%  %\vspace*{0.2cm}
%  \item PEP solution is second order accurate (exact mean and standard deviation)
% 
%  %\vspace*{0.2cm}
%  \item SEPFEM solution (PDFs) can be made as accurate as need be
% 
% 
%  \item Tails of PDFs can than be used to develop accurate risk
% 
% 
%  \item Application to a realistic case of seismic wave propagation
%\vspace*{0.2truecm}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_PlotEd.pdf}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_PDFedited.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic FEM}
\begin{itemize}
\item Minimizing norm of error of finite representation using Galerkin
technique (Ghanem and Spanos 2003):
\vspace*{0.6truecm}
\begin{flushright}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn}^{ep} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K_{mnk}^{'ep} = \left< F_m \psi_i[\{\xi_r\}] \right >
\end{equation}
\end{flushright}
% \begin{itemize}
%
% \vspace*{0.5cm}
% \item Final eqn.:
%
% \vspace*{0.4cm}
% \begin{flushright}
% \begin{normalsize}
% \begin{equation}
% \nonumber
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\zeta_r\}] \right >
% \end{equation}
% \end{normalsize}
% \end{flushright}
\vspace*{0.5cm}
\begin{equation}
\nonumber
K_{mn}^{ep} = \int_D B_n \textcolor{mycolor}{E}^{ep} B_m dV
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
K_{mnk}^{'ep} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
\end{equation}
\vspace*{1.0cm}
\begin{equation}
\nonumber
C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right >
\ \ \ \ \ \ \ \ \ \ \ \
F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
%\item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
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\subsection{Seismic Wave Propagation Through Uncertain Soils}
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\begin{frame}
\frametitle{"Uniform" CPT Site Data}
%\vspace*{0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
\end{center}
%\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Full PDFs of all DOFs (and $\sigma_{ij}$, $\epsilon_{ij}$, etc.)}
%\frametitle{Full PDFs for Real Data Case}
\begin{itemize}
\vspace*{0.7cm}
\item Stochastic ElasticPlastic\\
Finite Element Method \\
(SEPFEM) \\
\vspace*{0.5cm}
\item Dynamic case
\vspace*{0.5cm}
\item Full PDF at \\
each time step $\Delta t$
\end{itemize}
\vspace*{4.60cm}
\begin{flushright}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.pdf}
%\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{flushright}
%
\end{frame}
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\begin{frame}
\frametitle{PDF at each $\Delta t$ (say at $6$ s)}
\begin{figure}
\begin{center}
\hspace*{1.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{PDF $\rightarrow$ CDF (Fragility) at $6$ s}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Probability of Unacceptable Deformation ($50$cm)}
\begin{figure}
\begin{center}
\vspace*{0.3cm}
%\hspace*{0.75cm}
\includegraphics[width=10.50cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrietaEdited_ps.pdf}
\vspace*{0.5cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
\end{frame}
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\section{Summary}
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%
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\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item High fidelity, time domain, nonlinear, earthquake soil structure
interaction (ESSI) modeling and simulations (deterministic and probabilistic)
\vspace*{0.1cm}
\item The ESSI Simulator System (Program, Computer, Lecture Notes)
\vspace*{0.1cm}
\item Educational effort is essential (USNRC, CNSC, IAEA, UCD, LBNL, INL,
companies), seminars, short courses
%\vspace*{0.1cm}
%\item Information Portal:\\
%{\large \bf
%\url{http://nrcessisimulator.info}}
\vspace*{0.1cm}
\item Funding from the USNRC, DOE, NSF, and CNSC is much appreciated
\end{itemize}
\end{frame}
\end{document}